arXiv subjects

Yemon Choi

Publications and source records attributed to Yemon Choi.

At least 37 records · Page 2Linked to original sources

arXiv2013

A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra

It has been a longstanding problem whether every amenable operator algebra is isomorphic to a (necessarily nuclear) C*-algebra. In this note, we give a nonseparable counterexample. The existence of a separable counterexample remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in C*-algebras and show that our method cannot produce a separable counterexample.

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arXiv2013

ZL-amenability and characters for the restricted direct products of finite groups

Let $G$ be a restricted direct product of finite groups $\{G_i \}_{i\in I}$, and let $\Zl^1(G)$ denote the centre of its group algebra. We show that $\Zl^1(G)$ is amenable if and only if $G_i$ is abelian for all but finitely many $i$, and characterize the maximal ideals of $\Zl^1(G)$ which have bounded approximate identities. We also study when an algebra character of $\Zl^1(G)$ belongs to $c_0$ or $\ell^p$ and provide a variety of examples.

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arXiv2013

ZL-amenability constants of finite groups with two character degrees

We calculate the exact amenability constant of the centre of $\ell^1(G)$ when $G$ is one of the following classes of finite group: dihedral; extraspecial; or Frobenius with abelian complement and kernel. This is done using a formula which applies to all finite groups with two character degrees. In passing, we answer in the negative a question raised in work of the third author with Azimifard and Spronk (J. Funct. Anal. 2009).

math.FA↗

arXiv2013

Approximately multiplicative maps from weighted semilattice algebras

We investigate which weighted convolution algebras $\ell^1_ω(S)$, where $S$ is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all $\ell^1_ω(S)$ where $S$ has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein (IJMMS, 1999). We also investigate when $(\ell^1_ω(S),{\bf M}_2)$ is an AMNM pair in the sense of Johnson (JLMS, 1988), where ${\bf M}_2$ denotes the algebra of 2-by-2 complex matrices. In particular, we obtain the following two contrasting results: (i) for many non-trivial weights on the totally ordered semilattice ${\bf N}_{\min}$, the pair $(\ell^1_ω({\bf N}_{\min}),{\bf M}_2)$ is not AMNM; (ii) for any semilattice $S$, the pair $(\ell^1(S),{\bf M}_2)$ is AMNM. The latter result requires a detailed analysis of approximately commuting, approximately idempotent $2\times 2$ matrices.

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arXiv2013

Quotients of Fourier algebras, and representations which are not completely bounded

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.

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arXiv2012

Singly generated operator algebras satisfying weakened versions of amenability

We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability.

math.OA↗

arXiv2011

On commutative, operator amenable subalgebras of finite von Neumann algebras

An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.

math.OA↗

arXiv2011

Simplicial cohomology of band semigroup algebras

We establish simplicial triviality of the convolution algebra $\ell^1(S)$, where $S$ is a band semigroup. This generalizes results of the first author [Glasgow Math. J. 2005, Houston J. Math. 2010]. To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on $\ell^1(S)$. Crucial to our approach is the use of the structure semilattice of $S$, and the associated grading of $S$, together with an inductive normalization procedure in cyclic cohomology; the latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.

math.FA↗

arXiv2010

A note on some group $C^*$-algebras which are quasi-directly finite

An algebra is said to be quasi-directly finite when any left-invertible element in its unitization is automatically right-invertible. It is an old observation of Kaplansky that the von Neumann algebra of a discrete group has this property; in this note, we collate some analogous results for the group $C^*$-algebras of more general locally compact groups. Partial motivation comes from earlier work of the author on the phenomenon of empty residual spectrum for convolution operators.

math.OA↗

arXiv2010

Splitting maps and norm bounds for the cyclic cohomology of biflat Banach algebras

We revisit the old result that biflat Banach algebras have the same cyclic cohomology as $\mathbb C$, and obtain a quantitative variant (which is needed in forthcoming joint work of the author). Our approach does not rely on the Connes-Tsygan exact sequence, but is motivated strongly by its construction as found in [Connes,1985] and [Helemskii,1992].

math.KT↗

arXiv2010

Characterising derivations from the disc algebra to its dual

We show that the space of all bounded derivations from the disc algebra into its dual can be identified with the Hardy space $H^1$; using this, we infer that all such derivations are compact. Also, given a fixed derivation $D$, we construct a finite, positive Borel measure $μ_D$ on the closed disc, such that $D$ factors through $L^2(μ_D)$. Such a measure is known to exist, for any bounded linear map from the disc algebra to its dual, by results of Bourgain and Pietsch, but these results are highly non-constructive.

math.FA↗

arXiv2010

Translation-finite sets, and weakly compact derivations from $\lp{1}(\Z_+)$ to its dual

We characterize those derivations from the convolution algebra $\ell^1({\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of "translation-finite" subsets of ${\mathbb Z}_+$, and we investigate how this notion relates to other notions of "smallness" for infinite subsets of ${\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.

math.FA↗

arXiv2010

Group representations with empty residual spectrum

Let $X$ be a Banach space on which a discrete group $Γ$ acts by isometries. For certain natural choices of $X$, every element of the group algebra, when regarded as an operator on $X$, has empty residual spectrum. We show, for instance, that this occurs if $X$ is $\ell^2(\Gm)$ or the group von Neumann algebra $VN(\Gm)$. In our approach, we introduce the notion of a {\em surjunctive pair}, and develop some of the basic properties of this construction. The cases $X=\ell^p(\Gm)$ for $1<p<2$ or $2<p<\infty$ are more difficult. If $\Gm$ is amenable we can obtain partial results, using a majorization result of Herz; an example of Willis shows that some condition on $\Gm$ is necessary.

math.FA↗

arXiv2009

Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras

Amenability of any of the algebras described in the title is known to force them to be finite-dimensional. The analogous problems for \emph{approximate} amenability have been open for some years now. In this article we give a complete solution for the first two classes, using a new criterion for showing that certain Banach algebras without bounded approximate identities cannot be approximately amenable. The method also provides a unified approach to existing non-approximate amenability results, and is applied to the study of certain commutative Segal algebras. Using different techniques, we prove that \emph{bounded} approximate amenability of the second dual of a Fourier algebra implies that it is finite-dimensional. Some other results for related algebras are obtained.

math.FA↗

arXiv2009

Uniform bounds for point cohomology of $\ell^1({\mathbb Z}_+)$ and related algebras

It is well-known that the point cohomology of the convolution algebra $\ell^1({\mathbb Z}_+)$ vanishes in degrees 2 and above. We sharpen this result by obtaining splitting maps whose norms are bounded independently of the choice of point module. Our construction is a by-product of new estimates on projectivity constants of maximal ideals in $\ell^1({\mathbb Z}_+)$. Analogous results are obtained for some other $L^1$-algebras which arise from `rank one' subsemigroups of ${\mathbb R}_+$.

math.FA↗

arXiv2008

Hochschild homology and cohomology of {\ell}^1({\mathbb Z}_+^k)

Building on the recent determination of the simplicial cohomology groups of the convolution algebra ${\ell}^1({\mathbb Z}_+^k)$ [Gourdeau, Lykova, White, 2005] we investigate what can be said for cohomology of this algebra with more general symmetric coefficients. Our approach leads us to a discussion of Harrison homology and cohomology in the context of Banach algebras, and a development of some of its basic features. As an application of our techniques we reprove some known results on second-degree cohomology.

math.FA↗

arXiv2008

Simplicial cohomology of augmentation ideals in ${\ell}^1(G)$

Let $G$ be a discrete group. We give a decomposition theorem for the Hochschild cohomology of $\ell^1(G)$ with coefficients in certain $G$-modules. Using this we show that if $G$ is commutative-transitive, the canonical inclusion of bounded cohomology of $G$ into simplicial cohomology of $\ell^1(G)$ is an isomorphism.

math.FA↗

arXiv2008

Simplicial homology of strong semilattices of Banach algebras

Certain semigroups are known to admit a `strong semilattice decomposition' into simpler pieces. We introduce a class of Banach algebras that generalise the $\ell^1$-convolution algebras of such semigroups, and obtain a disintegration theorem for their simplicial homology. Using this we show that for any Clifford semigroup $S$ of amenable groups, $\lp{1}(S)$ is simplicially trivial: this generalises results in \cite{YC_GMJ}. Some other applications are presented.

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